Bending of plates

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Bending of plates or plate bending refers to the deflection of a plate perpendicular to the plane of the plate under the action The resulting displacement is The amount of deflection can be determined by solving the differential equations of an of external forces and moments. appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

Bending of an edge clamped circular plate under the action of a transverse pressure. The left half of the plate shows the deformed shape while the right half shows the undeformed shape. This calculation was performed using Ansys.

Bending of Kirchhoff-Love plates
In the Kirchhoff–Love plate theory for plates the governing equations are[1]

and

In expanded form,

and

Forces and moments on a flat plate.

where and

is an applied transverse load per unit area, the thickness of the plate is

, the stresses are

,

The quantity

has units of force per unit thickness. The quantity

has units of moment per unit thickness. these equations reduce to[2]

For isotropic, homogeneous, plates with Young's modulus

and Poisson's ratio

where

is the deflection of the mid-surface of the plate.

In rectangular Cartesian coordinates,

Circular Kirchhoff-Love plates
The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions. These solutions were first found by Poisson in 1829. Cylindrical coordinates are convenient for such problems. The stresses are The governing equation in coordinate-free form is

Cylindrical plate bending
Cylindrical bending occurs when a rectangular plate that has dimensions , where perpendicular to the plane of the plate. In cylindrical coordinates , Such a plate takes the shape of the surface of a cylinder. Simply supported plate with axially fixed ends For a simply supported plate under cylindrical bending with edges that are free to rotate but have a fixed Levy techniques. For symmetrically loaded circular plates, , and we have . Cylindrical bending solutions can be found using the Navier and and the thickness is small, is subjected to a uniform distributed load

Bending of thick Mindlin plates
For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation. Mindlin's theory provides one approach for find the deformation and stresses in such plates. Solutions to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using canonical relations.[3] Therefore, the governing equation is

Governing equations
The canonical governing equation for isotropic thick plates can be expressed as[3] If and are constant, direct integration of the governing equation gives us

where

are constants. The slope of the deflection surface is is the bending rigidity, is the plate thickness, , is

where is the applied transverse load, is the shear modulus, the shear correction factor, is the Young's modulus, is the Poisson's ratio, and

For a circular plate, the requirement that the deflection and the slope of the deflection are finite at

implies that

.

Clamped edges
In Mindlin's theory, is the transverse displacement of the mid-surface of the plate and the quantities and are the rotations of the mid-surface normal about the -axes, respectively. The canonical parameters are and . The shear correction has the value For a circular plate with clamped edges, we havefor this theory and at the edge of the plate (radius factor ). Using usually these boundary conditions. we get The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations and

The in-plane displacements in the plate are

The in-plane strains in the plate are where is the displacement predicted for a Kirchhoff-Love plate, , and is a biharmonic function such that , is a function that satisfies the Laplace equation,

vanish, and the Mindlin solution is related to the corresponding Kirchhoff solution by

The moment resultants (bending moments) are

Bending Reissner-Stein cantilever plates The maximumof radial stress is at and :
Reissner-Stein theory for cantilever plates[4] leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load at .

where

. The bending moments at the boundary and the center of the plate are

and the boundary conditions at

are

Rectangular Kirchhoff-Love plates
For rectangular plates, Navier in 1820 introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and Solution of this system ofthen two superimpose ODEs gives the Fourier components to get the solution for an arbitrary load.

Sinusoidal load
Let us assume that the load is of the form

where Here is the amplitude,

. The bending moments and shear forces corresponding to the displacement is the width of the plate in the -direction, and is the width of the plate in the -direction. along the edges of the plate is zero, the bending moment .

are is

Since the plate is simply supported, the displacement zero at and , and is zero at

and

If we apply these boundary conditions and solve the plate equation, we get the solution

We can calculate the stresses and strains in the plate once we know the displacement. For a more general load of the form
Bending of a rectangular plate under the action of a distributed force per unit area.

where

and

are integers, we get the solution

The stresses are

Navier solution
Let us now consider a more general load . We can break this load up into a sum of Fourier components such that If the applied load at the edge is constant, we recover the solutions for a beam under a concentrated end load. If the applied load is a linear function of , then

where

is an amplitude. We can use the orthogonality of Fourier components,

See also
Bending Infinitesimal strain theory Kirchhoff–Love plate theory to findLinear the amplitudes . Thus we have, by integrating over , elasticity Mindlin–Reissner plate theory Plate theory Stress (mechanics) Structural acoustics Vibration of plates If we repeat the process by integrating over , we have

Retrieved from "http://en.wikipedia.org/w/index.php?title=Bending_of_plates&oldid=540507140" Categories: Continuum mechanics Now that we know , we can just superpose solutions of the form given in equation (1) to get the displacement, i.e., This page was last modified on 26 February 2013 at 05:39. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of Use for details. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

Uniform load Consider the situation where a uniform load is applied on the plate, i.e., . Then

Now

We can use these relations to get a simpler expression for

:

Since are odd:

[ so

] when

and

are even, we can get an even simpler expression for

when both

and

Plugging this expression into equation (2) and keeping in mind that only odd terms contribute to the displacement, we have

The corresponding moments are given by

The stresses in the plate are

Displacement (

)

Stress (

)

Stress (

) for a rectangular plate with mm, mm, mm, GPa, and under a load kPa.

Displacement and stresses along

The red line represents the bottom of the plate, the green line the middle, and the blue line the top of the plate.

Levy solution
Another approach was proposed by Levy in 1899. In this case we start with an assumed form of the displacement and try to fit the parameters so that the governing equation and the boundary conditions are satisfied. Let us assume that

For a plate that is simply supported at and (verify). The goal is to find . Moments along edges

, the boundary conditions are and such that it satisfies the boundary conditions at

. The moment boundary condition is equivalent to and and, of course, the governing equation

Let us consider the case of pure moment loading. In that case the governing equation can be expanded as

and

has to satisfy

. Since we are working in rectangular Cartesian coordinates,

Plugging the expression for

in the governing equation gives us

or

This is an ordinary differential equation which has the general solution

where

are constants that can be determined from the boundary conditions. Therefore the displacement solution has the form

Let us choose the coordinate system such that the boundaries of the plate are at moment boundary conditions at the boundaries are

and

(same as before) and at

(and not

and

). Then the

where

are known functions. The solution can be found by applying these boundary conditions. We can show that for the symmetrical case where

and

we have

where

Similarly, for the antisymmetrical case where

we have

We can superpose the symmetric and antisymmetric solutions to get more general solutions. Uniform and symmetric moment load For the special case where the loading is symmetric and the moment is uniform, we have at ,

Displacement (

)

Bending stress (

)

Transverse shear stress (

) and . The bending stress is along the bottom surface of the plate. The

Displacement and stresses for a rectangular plate under uniform bending moment along the edges transverse shear stress is along the mid-surface of the plate.